Ta có:
\(\dfrac{\sqrt[4]{17+12\sqrt{2}} +\sqrt[4]{17-12\sqrt{2}}}{2}\)
\(=\dfrac{\sqrt[4]{3^2+2.3.(2\sqrt{2})+\left(2\sqrt{2}\right)^2}+\sqrt[4]{3^2-2.3.(2\sqrt{2})+\left(2\sqrt{2}\right)^2}}{2}\)
\(=\dfrac{\sqrt[4]{\left(3+2\sqrt{2}\right)^2}+\sqrt[4]{\left(3-2\sqrt{2}\right)^2}}{2}\)
\(=\dfrac{\sqrt[4]{\left(2+2\sqrt{2}+1\right)^2}+\sqrt[4]{\left(2-2\sqrt{2}+1\right)^2}}{2}\)
\(=\dfrac{\sqrt[4]{[\left(\sqrt{2}+1\right)^2]^2}+\sqrt[4]{[\left(\sqrt{2}-1\right)^2]^2}}{2}\)
\(=\dfrac{\sqrt[4]{\left(\sqrt{2}+1\right)^4}+\sqrt[4]{\left(\sqrt{2}-1\right)^4}}{2}\)
\(=\dfrac{\sqrt{2}+1+\sqrt{2}-1}{2}\)
\(=\dfrac{2\sqrt{2}}{2}\)
\(=\sqrt{2}\) (đpcm)
